The transmutation circles in Fullmetal Alchemist always looked familiar to me, but I could never place my finger on why. What, if anything, are they based on?
This site has a note about "Squaring the circle", describing a utilization of circles:
Attributed to Michael Maier, a German alchemist from the 17th century. That gives you the generalized "simple" transmutation circle:
The symbols used in some of the more complex circles are old alchemic symbols or variations of them. I don't know of a complete list of them but some of them can be seen in Annibal Barlet's works or other symbols from the 17th century. The inscriptions themselves seem to mostly come from George Ripley's 12 Gates.
Circles are a relatively major thing in alchemy. Ouroboros symbols were sigils in alchemy dating back to Cleopatra, and they depict a serpent or dragon eating its own tail.
The symbols in the circle are mostly alchemic symbols. The one on the the bottom left on the outside looks like the symbol for iron ore or the symbol for male. Going clockwise from that is what looks like either copper ore or female, a symbol I can't identify, one of the symbols for regulus, another symbol I can't identify, and sal-ammoniac. (Source)
Some Renaissance alchemic images have circles with words in them.
Squaring the circle was another major part of Renaissance alchemic ideas. It was believed that this symbol plus a man and a woman in the circle was all that was needed to create a philosopher's stone. A full explanation can be seen here.
The Seal of Solomon is another similar image.
According to here:
In alchemy, the combination of the fire and water symbols (up and down triangles) is known as the Seal of Solomon. The symbol is representative of the combination of opposites and transmutation. By combining the alchemical symbols for fire (upwards triangle) and water (downwards triangle), the alchemical symbols for earth and air are also created. The downwards facing triangle is divided along the center by the base line of the opposite triangle. This is the alchemical symbol for earth. Conversely, the upwards triangle divided by the base line of the downwards triangle is the alchemical symbol for air. The Seal of Solomon is all that is unified in perfect balance; the Spirit Wheel.
It's seems to be pretty much a compilation of different alchemic elements.
Yes, those are the symbols for the four base elements. But there are also two more very important parts to the circle. The center point which represents origin, time and growth, as well as an outer circle that represents the womb, the ether and protection. There are many more aspects and details to Solomon's circle than that. In fact, for such a simple symbol it is quite complex.